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We calculate the double-virtual $\eta \rightarrow \gamma^* \gamma^*$transition form factor $\mathcal{F}_{\eta \to \gamma^* \gamma^*}(q_1^2,q_2^2)$from first principles using a lattice QCD simulation with $N_f=2+1+1$ quarkflavors at the physical pion mass and at one lattice spacing and volume. Thekinematic range covered by our calculation is complementary to the oneaccessible from experiment and is relevant for the $\eta$-pole contribution tothe hadronic light-by-light scattering in the anomalous magnetic moment $a_\mu= (g-2)/2$ of the muon. From the form factor calculation we extract the partialdecay width $\Gamma(\eta \rightarrow \gamma \gamma) =323(85)_\text{stat}(22)_\text{syst}$ eV and the slope parameter$b_\eta=1.19(36)_\text{stat}(16)_\text{syst}$ GeV${}^{-2}$. For the $\eta$-polecontribution to $a_\mu$ we obtain $a_\mu^{\eta-\text{pole}} =13.2(5.2)_\text{stat}(1.3)_\text{syst} \cdot 10^{-11}$.

physical review. d/physical review. d.

The $\eta \rightarrow \gamma^* \gamma^*$ transition form factor and the  hadronic light-by-light $\eta$-pole contribution to the muon $g-2$ from  lattice QCD